In this article we prove a Wiener Tauberian theorem for L^p(SL_2(R)), 1\leq p <2. Let G be the group SL_2(R) and K its maximal compact subgroup SO(2,R). Let M be {\pm I}. We show that if the Fourier transforms of a set of functions in L^p(G) do not vanish simultaneously on any irreducible L^{p-\epsilon}-tempered representation for some \epsilon>0, where they are assumed to be defined, and if for each M-type at least one of the matrix coefficients of any of those Fourier transforms does not `decay too rapidly at infinity' in a certain sense, then this set of functions generate L^p(G) as a L^1(G)-bimodule. As a key step towards this main theorem we prove a W-T Theorem for L^p-sections of certain line bundles over G/K. W-T theorems on SL_2(R) have been proved so far, for biinvariant L^1 functions and for L^1 functions on the symmetric space SL_2(R)/SO(2,R), where the generator is left K-finite. Our results are on the space of all L^p functions (resp. sections), p\in[1, 2) of SL_2(R) (resp. of line bundles over SL_2(R)/SO(2,R)), without any restriction of K-finiteness on the generators.